Let $(W_s)_{s\geq 0}$ be a Brownian motion and $t$ a fixed point in time. What is the distribution of $$\Big.\int_0^tW_sds\Big|W_t$$ i.e. the integral of a Brownian bridge with respect to time? Is it Gaussian?
Asked
Active
Viewed 1,619 times
1 Answers
1
For convenience, I will take $t=1$. The same kind of calculations apply for general $t$. As, $W_s\vert W_1$ is Gaussian process, the lebesgue integral is Gaussian RV : Approximate the given integral as Riemann sums and each Riemann sum is Gaussian and hence the limit will be Gaussian. Look here for similar problem : click here
The expectation of it is given by $\int_0^1tW_1dt=0.5W_1$
chandu1729
- 3,801
-
2Actually the expectation is $\frac12tW_t$, thus, $\frac12W_1$ assuming that $t=1$. – Did Dec 25 '14 at 10:20
-
@Did : Edited the answer. Thanks! – chandu1729 Dec 25 '14 at 16:38
-
yeah but what is its variance? can't find it anywhere – carlo Sep 12 '19 at 13:00